Question

7. [3 marks] Find the

following sums
7. [3 marks] Find the following sums \[ \sum_{n=0}^{\infty} \frac{3 \times 5^{n}}{3^{2 n}} \]

Answer 1

`\displaystyle (27)/(4)`

Explanation:

Rewrite infinite series

`\displaystyle \sum_(n=0)^(\infty) (3*5^(n))/(3^(2 n))=\sum_(n=0)^(\infty) (3*5^(n))/(9^n)=\sum_(n=0)^(\infty) 3\biggr((5)/(9)\biggr)^n`

Since we have a common ratio of
`r=(5)/(9)` and the first term is
`a_1=3`, then we can get the sum of the infinite geometric series:

`\displaystyle S_n=(a_1)/(1-r)=(3)/(1-(5)/(9))=(3)/((4)/(9))=3*(9)/(4)=(27)/(4)`